Environmental Engineering Reference
In-Depth Information
9.4 MOTION IN A CENTRAL FORCE
Equation (9.35) certainly constitutes progress in solving for the general motion of
two isolated bodies. We shall now make two further assumptions which will allow
us to solve the problem completely. Firstly, we assume that the force is 'central',
which means that it acts along the line joining the particles, i.e.
F
e
r
. Secondly,
we assume that the force is conservative which, for a central force, means that it
depends only on the distance between the particles and not on their orientation, i.e.
F
∝
∝
f(r)
e
r
where
f(r)
is related to the corresponding potential via
d
U
d
r
=
f(r).
These are not very restrictive assumptions and the gravitational interaction between
two particles satisfies them both.
Our task is to solve
d
U
d
r
e
r
µ
r
¨
=−
(9.37)
which is still a system of three coupled second order differential equations. Now,
since the force is conservative we know that the sum of the kinetic and potential
energies must be conserved. In addition, the central nature of the force leads also
to the conservation of angular momentum, as we shall now show.
The total angular momentum of the system about some origin is
L
=
m
1
x
1
× ˙
x
1
+
m
2
x
2
× ˙
x
2
.
(9.38)
Substituting using Eq. (9.34) and choosing to work in the centre-of-mass frame
(i.e.
R
=
0
) implies that
×
˙
L
=
µ
r
r
.
(9.39)
This is easily seen to be a constant vector since
d
L
d
t
=
µ
r
×
r
+
µ
r
×
r
=
0
(9.40)
and we used the fact that
r
is parallel to
r
for a central force. Outside of the
centre-of-mass frame, angular momentum is of course still conserved (this is just a
result of the central nature of the force) but it is only in that frame that the angular
momentum takes on the form written in Eq. (9.39). The existence of these two
conserved quantities will help us greatly.
Indeed, the conservation of angular momentum implies immediately that the
motion must be planar. Generally speaking the position vector
r
is constrained
always to lie in the plane which is perpendicular to the angular momentum vector.
This follows immediately from Eq. (9.39) since the nature of the vector product
implies that
L
is always perpendicular to the position vector
r
. Now since the
¨
